On representativeness of the representative cellsfor the microstructure-based predictions ofsound absorption in fibrous and porous media

Tomasz G. ZielińskiDepartment of Intelligent Technologies, Institute of Fundamental Technological Research of the PolishAcademy of Sciences, ul. Pawinskiego 5B, 02-106 Warsaw, Poland. E-mail: tzielins@ippt.pan.pl

SummaryRealistic microstructure-based calculations have recently become an important tool for a performanceprediction of sound absorbing porous media, seemingly suitable also for a design and optimization ofnovel acoustic materials. However, the accuracy of such calculations strongly depends on a correctchoice of the representative microstructural geometry of porous media, and that choice is constrainedby some requirements, like, the periodicity, a relative simplicity, and the size small enough to allowfor the so-called separation of scales. This paper discusses some issues concerning this important mat-ter of the representativeness of representative geometries (two-dimensional cells or three-dimensionalvolume elements) for sound absorbing porous and fibrous media with rigid frame. To this end, theaccuracy of two- and three-dimensional cells for fibrous materials is compared, and the microstructure-based predictions of sound absorption are validated experimentally in case of a fibrous material madeup of a copper wire. Similarly, the numerical predictions of sound absorption obtained from some reg-ular Representative Volume Elements proposed for porous media made up of loosely-packed identicalrigid spheres are confronted with the corresponding analytical estimations and experimental results.Finally, a method for controlled random generation of representative microstructural geometries forsound absorbing open foams with spherical pores is briefly presented.

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1. Introduction

Sound absorption in air-saturated rigid porous mediacan be very well predicted by the Johnson-Champoux-Allard model [1] or its enhanced versions, providedthat all model parameters are known. Those parame-ters can be divided into two groups: the inherent pa-rameters of air and the so-called transport parame-ters which depend purely on the micro-geometry ofthe solid frame (skeleton) of porous medium.

A microstructure-based approach has been devel-oped for the problem of the macroscopic propagationof acoustic waves in a porous medium, where the esti-mations of the transport parameters are derived fromthe micro-geometry of the rigid solid frame (skeleton).This approach has been recently applied to variousporous and fibrous materials [2, 3, 4, 5, 6, 7, 8]. In thisapproach, of the utmost importance are the periodicRepresenative Volume Elements (RVEs) and a ques-tion of their representativeness for particular porous

(c) European Acoustics Association

media. The present work discusses the microstructuralapproach and the question of RVE for a fibrous mate-rial made up of a metal wire, a granular layer of plasticspherical beads, and a foam with spherical pores.

2. Sound absorption of porous media

2.1. Experimental testing

Sound absorption of porous media can be measured inthe impedance tube (see Figure 1) using the so-calledtwo-microphone transfer function method [9]. A sam-ple of porous material of known thickness is set at therigid termination at one end of tube. A loudspeaker,mounted at the other end, generates plane harmonicacoustic waves which propagate in the tube, pene-trate the sample, and are reflected: partially from thesurface of sample, and completely from the rigid ter-mination at the back of the sample. A standing-waveinterference pattern results due to the superpositionof forward- and backward-travelling waves. By mea-suring acoustic pressure at two distinct positions us-ing two calibrated microphones, the transfer functionis determined, which allows to calculate the surface

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Figure 1. Impedance tube and a fibrous or granular sampleinside the tube.

acoustic impedance at the sample’s surface, and thenthe reflection and absorption coefficients.

In this paper, the impedance tube measurementswill be confronted with the results of modelling basedon representative microstructures for two porous me-dia (see Figure 1): a fibrous material woven from acopper wire and a granular layer composed of plasticspherical beads.

2.2. Microstructural modelling

The one-dimensional configuration of porous layer ofknown thickness (i.e., the sample in the tube) sub-jected to harmonic excitation by an acoustic wave canbe solved analytically using the Helmholtz equation,provided that the solid frame of porous medium isstiff and motionless (rigid), the porosity is open, andfinally, the (effective) speed of sound ce in such porousmaterial is known. It is in fact, an effective complex-valued function of frequency (to reflect the dispersiveand absorptive properties of porous medium), whichcan be related to the effective density %e and effectivebulk modulus Ke of porous medium in the standardway, namely: ce =

√Ke/%e. Those are also complex

and frequency-dependent functions, and they are re-lated to the constant density %f and bulk modulus Kfof the actual fluid filling the porous medium (typi-cally, it is the air) in the following way:

%e(ω) =%f α(ω)

φ, Ke(ω) =

Kf

φβ(ω),

where ω is the angular frequency, φ is the open poros-ity, and

β(ω) = γf −γf − 1

α′(ω).

Here, γf is the ratio of specific heats for thefluid in pores (air), whereas α(ω) and α′(ω) arecomplex-valued frequency dependent functions de-scribing the visco-thermal interaction between the

fluid in pores (air) and the solid skeleton (rigid frame)of porous medium. The Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model provides analytical for-mulas to estimate these functions.

The function α(ω) is called the dynamic (viso-inertial) tortuosity function and the formula for it de-pends on the kinematic viscosity of air (fluid in pores)and some purely geometric parameters of solid frame,namely: the total open porosity, the (classic, iner-tial) tortuosity, the static viscous tortuosity, the static(viscous) permeability, and the characteristic lengthfor viscous forces. The function α′(ω) is the thermalanalogue of dynamic tortuosity and its JCAPL-modelformula contains the kinematic viscosity and Prandtlnumber of air (fluid in pores) and the following geo-metric parameters of solid frame: the total porosity,the static thermal tortuosity, the static thermal ana-logue of permeability, and the characteristic length forviscous effects. Obviously, both formulas depend alsoon the angular frequency ω.

Thus, there are eight purely geometric parametersin the JCAPL model, the so-called transport param-eters of porous medium, namely: the porosity, thetortuosity, the static viscous and thermal tortuosi-ties, the viscous and thermal permeabilities, and thevisocus and thermal characteristic lengths. Providedthat there exists a periodic cell representative for themicro-geometry of porous medium, the transport pa-rameters can be calculated directly from microstruc-ture by solving three static analyses defined on thefluid domain of periodic cell, namely:• the Stokes flow (i.e., the steady viscous flow) caused

by a unit pressure gradient constant throughout thefluid domain, with no-slip boundary conditions onthe fluid-solid interface;

• the steady heat transfer caused by a unit heatsource constant in the fluid domain, with isother-mal boundary conditions on the fluid-solid inter-face;

• the Laplace problem (the potential flow) with aunit source constant in the fluid domain.

In fact, two of the transport parameters (the poros-ity and thermal length) are determined directly fromthe micro-geometry. The remaining six parameters arefound by averaging (over the cell volume) and rescal-ing the fields found by solving the problems listedabove. This approach will be applied below for threevarious rigid porous media: a fibrous material of cop-per wire, a granular layer of spherical beads, and arandomly generated microstructure of open foam.

3. Examples of absorption estimationusing various representative cells

3.1. Fibrous material

A silvered copper wire with diameter 0.5mm was usedto manufacture two fibrous samples by weaving and

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Figure 2. Representative Volume Element (RVE) for a fi-brous material with porosity 90%.

fitting pieces of the wire into the impedance tube of di-ameter 29mm (see Figure 1). Two cylindrical sampleswere manufactured in that way: one is 30mm in heightand the other 60mm. The smaller one was made upfrom 10m and the taller one from 20m of copper wire,so that both samples have the same porosity of 90%.

The sound absorption was measured in theimpedance tube, in the frequency range from 100Hzto 6 kHz, for both samples separately, and also fortwo-layered configurations with total height 90mm,composed from the fibrous samples set closely oneafter another, that is, one sample in front and theother behind (i.e., 30mm+60mm), and conversely(i.e., 60mm+30mm). The acoustic absorption coef-ficient was also computed form the microstructuralmodelling where the Representative Volume Elementpresented in Figure 2 was used. The RVE is periodicand contains only straight fibres which are mainlygrouped in a layer normal to the direction of prop-agation X. In a simplified way, such a configurationto some point resembles the distributions of locallystraight fibres in the real samples.

The results of measurements and calculations arepresented together in Figure 3. One should notice thattwo experimental curves obtained for the two-layeredconfigurations of the same total height of 90mm arenearly identical, which proves that the same fibrousmaterial is always manufactured, although the pro-cedure involves manual weaving of copper wire. Oneshould also observe that the computed absorptioncurves are very similar to their experimental coun-terparts which means that the proposed RVE – basedon the assumption that the fibres are locally straight– is in fact representative.

3.2. Granular medium

An artificial granular medium was composed fromidentical plastic beads with diameter of 5.9mm: thebeads were loosely poured into the impedance tube(with diameter of 29mm, see Figure 1) set up verti-cally, so that a layer of beads, 106mm thick and witha roughly flat surface, was formed. The acoustic ab-sorption coefficient was measured for the layer in thefrequency range from 500Hz to 6 kHz (see Figure 6).

Figure 4. Simple Cubic (SC) arrangement of slightly over-lapping spheres and the corresponding RVE of fluid do-main.

Figure 5. Face Centered Cubic (FCC) arrangement ofspheres slightly shifted apart and the corresponding RVEof fluid domain.

Since all the beads in the granular layer are identicalwith the same size and the same weight of 0.3 g, thelayer’s porosity could be easily determined by simplymeasuring the total weight of layer, so that the num-ber of solid beads in layer could be estimated as 380,which gives the total porosity of approximately 42%.

The microstructre-based approach was applied forgranular media of that kind by Gasser et al. [10], Leeet al. [11], and Zielinski [8, 12]. Some relevant an-alytical estimations were given by Boutin and Gein-dreau [13, 14]. Following the works by Zielinski [8, 12],two micro-geometries were constructed to representthe granular medium of plastic spherical beads. Bothare based on the well-known simple regular periodicarrangements of spheres, namely: the Simple Cubic(SC) arrangement which contains only one completesphere (composed from equal parts of eight spheres,see Figure 4), and the Face Centred Cubic (FCC) ar-rangement containing four complete spheres in the pe-riodic cube (see Figure 5). However, the SC arrange-ment has porosity of 47.6%, which is superior than theactual measured porosity, whereas the FCC arrange-ment has porosity of 26%, which is inferior. Therefore,in order to make these arrangements in some way rep-resentative, the porosity is set to the actual value of42% by letting the spheres to slightly overlap in theSC case, and shifting them slightly apart in the FCCcase. This can be clearly observed in the meshed RVEsof fluid domain presented in Figures 4 and 5.

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Acousticab

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Figure 3. Acoustic absorption of fibrous samples of tangled copper wire.

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Figure 6. Acoustic absorption of a 106mm thick porous layer of identical rigid beads (spheres).

The representative cells SC42% and FCC42% wereused by the finite element analyses in order to cal-culate the transport parameters from such repre-sentative microstructures. The calculated parametersserved in the JCAPL model and eventually the acous-tic absorption coefficient was computed for both casesfor the porous layer 106mm thick. These numericalresults are in Figure 6 compared with the experi-

mental curve. It is easy to notice that all curves aresimilar in character. In fact, the absorption peaks ofthe experimental curve are placed between the corre-sponding peaks obtained for the SC42% and FCC42%cases, which may be considered as two opposing limitcases. As a matter of fact, the results obtained for theFCC42% are closer to the experimental curve, which

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seems to be reasonable, since in this RVE the spheresare not overlapping.

3.3. Foam with spherical pores

An algorithm for random generation of periodic mi-crostructures for foams with spherical pores was pro-posed by Zielinski [15, 16]. The algorithm is based ona simulation of dynamic mixing of rigid spheres insidea shrinking cube of representative cell. In order to en-sure the periodicity of the representative cell, eachpore is represented by an assembly of eight spheresset in the vertices of a cube with the edge equal tothe current edge of the shrinking representative cell.Figure 7 illustrates this idea with three visualisationsof such random mixing of spheres – for legibility – in atwo-dimensional realisation of the procedure. A prac-tical final result of the three-dimensional realisationis also shown in Figure 7 in the form of some peri-odic arrangement of spheres representing five pores ina cubic cell.

In the proposed approach the generated random,periodic arrangements of pores are characterised by afew features, namely: the open porosity, the averagesize of pores and its standard deviation, and the typ-ical size of windows linking the pores. Except the lastone, those feature are directly controlled. The porosityis set as the stopping condition for the mixing algo-rithm (i.e., when the desired porosity is reached themixing and cube shrinking is accomplished), whereasthe required average size of pores is ensured by ap-plying an appropriate scaling for the final geometryof representative cell (i.e., in all finite element anal-yses the edge length is assumed in the way that en-sures the required average diameter of pores in thecell). The standard deviation of pores is set by choos-ing appropriate (initial) sizes of spheres. On the otherhand, the typical size of windows linking the poresis only indirectly controlled by the so-called maximalmutual penetration factor defined as ζ in Figure 8.By setting its value one approximately controls thesize of windows, since for two identical spheres (i.e.,R1 = R2 = Rp) the maximal widow radius Rw isrelated to the pore radius Rp in the following way:

Rw

Rp=

√ζ − ζ2

4.

Figure 9 shows a Representative Volume Element(see also the periodic assembly of overlapping spheresshown in Figure 7) for an open foam with porosity70%. The periodic representative cell contains fivespherical pores. The finite element mesh of the fluiddomain is also presented in Figure 9. It served inthe microstructural analyses to calculate the trans-port parameters for such porous medium. Then, thoseparameters were used by the JCAPL model to de-termine the effective density and speed of sound.Finally, the acoustic absorption coefficient was esti-mated for layers of such porous foam with various

Figure 7. Dynamic packing of periodic arrangements ofbubbles (pores) in a shrinking cell, and the final periodicarrangement of pores in a representative cube.

R1R2Rw

ζR1

ζR2

Figure 8. Two mutually penetrating spheres (pores).

Figure 9. Periodic skeleton of foam with porosity 70% andthe corresponding meshed fluid domain.

thickness, namely: 20mm, 30mm, and 40mm. Fig-ure 10 presents these results in the frequency rangefrom 200Hz to 5.2 kHz. The random distribution ofpores makes these estimations reliable provided thatthe average design features are fully representative forthe real foam.

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Figure 10. Acoustic absorption calculated for layers offoam with porosity 70% and various thickness

4. Conclusions

• More than a few pores, fibres or grains in a peri-odic cell (RVE) are necessary to well represent thegeometry of real porous or fibrous media.

• However, more pores (grains or fibres) in a rep-resentative cell means larger RVE and that wouldrequire more computational power.

• Moreover, the size of a large RVE may at higherfrequencies become comparable with the wave-lengths, which would worsen the accuracy and reli-ability of estimations, because of a weak separationof scales.

• A few (random) microstructures with the same av-erage features should be generated for one porousmedium and the estimations of acoustic wave prop-agation and absorption should be computed as anaverage result.

Acknowledgement

Financial support of Structural Funds in the Oper-ational Programme – Innovative Economy (IE OP)financed from the European Regional DevelopmentFund – Project “Modern Material Technologies inAerospace Industry” (No. POIG.01.01.02-00-015/08-00) – is gratefully acknowledged.

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